Homework 4
Exploratory Analysis
We begin by visualizing our data, and its distribution.
From this initial observation, we can see a general pattern of rainfall across the regions, as well as somewhat irregular normality patterns. However, to identify the best initial parameters to conduct our estimation, we must first plot an empirical variogram of each year of interest. And before that, we must first inspect the normality of our data.
## year= 1976 p value= 0.000001527713
## year= 1979 p value= 0.0000008326988
## year= 1982 p value= 0.0003342081We can see that our values are not normally distributed, after a thorough exploration we noticed that log transformations and square root transformations of our data improved the normality, since our estimation tools give us flexibility with the usage of lambda values we decided to include them in our process by automatically idntifying them usng the boxcoxfit function.
## variog: computing omnidirectional variogram## variog: computing omnidirectional variogram## variog: computing omnidirectional variogram
From the plots above we can see that the variograms become unstable after approx. 100 to 150 km. However, we must also identify a good limit that ensures isotropy. To do this, we perform a directional empirical variogram:
## variog: computing variogram for direction = 0 degrees (0 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing variogram for direction = 45 degrees (0.785 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing variogram for direction = 90 degrees (1.571 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing variogram for direction = 135 degrees (2.356 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing omnidirectional variogram## variog: computing variogram for direction = 0 degrees (0 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing variogram for direction = 45 degrees (0.785 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing variogram for direction = 90 degrees (1.571 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing variogram for direction = 135 degrees (2.356 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing omnidirectional variogram## variog: computing variogram for direction = 0 degrees (0 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing variogram for direction = 45 degrees (0.785 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing variogram for direction = 90 degrees (1.571 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing variogram for direction = 135 degrees (2.356 radians)
## tolerance angle = 22.5 degrees (0.393 radians)
## variog: computing omnidirectional variogram
These plots allow us to see the decay after 100 kms for all years except 1982, which seems to worsen much sooner. However, we assume that 100 km is a good enough cutting point for our estimations.
The following variograms will be the ones used for our fitting:
## variog: computing omnidirectional variogram## variog: computing omnidirectional variogram## variog: computing omnidirectional variogram
Estimation
From our variograms we can spot certain possible models that could fit our data; we attempt the Spherical, Exponential, Matérn (kappa=5), and Gaussian models.
Comment: The Gaussian model was not invertible; unsure as to why. Discuss with professor.
For each model, we cross-validate them and compare their root mean square error and coefficient of variation.
From these results, we can identify that the lowest error values correspond to the spherical model for the year 1976 and the exponential model for the remaining years.
## variog: computing omnidirectional variogram
## variofit: covariance model used is matern
## variofit: weights used: npairs
## variofit: minimisation function used: optim## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "0.01" "15.38" "0.01" "3"
## status "est" "est" "est" "fix"
## loss value: 0.00487828531209432
## ---------------------------------------------------------------
## likfit: likelihood maximisation using the function optim.
## likfit: Use control() to pass additional
## arguments for the maximisation function.
## For further details see documentation for optim.
## likfit: It is highly advisable to run this function several
## times with different initial values for the parameters.
## likfit: WARNING: This step can be time demanding!
## ---------------------------------------------------------------
## likfit: end of numerical maximisation.
## xvalid: number of data locations = 181
## xvalid: number of validation locations = 181
## xvalid: performing cross-validation at location ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181,
## xvalid: end of cross-validation## variog: computing omnidirectional variogram
## variofit: covariance model used is matern
## variofit: weights used: npairs
## variofit: minimisation function used: optim## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "2.04" "15.38" "0.68" "3"
## status "est" "est" "est" "fix"
## loss value: 406.120207235273
## ---------------------------------------------------------------
## likfit: likelihood maximisation using the function optim.
## likfit: Use control() to pass additional
## arguments for the maximisation function.
## For further details see documentation for optim.
## likfit: It is highly advisable to run this function several
## times with different initial values for the parameters.
## likfit: WARNING: This step can be time demanding!
## ---------------------------------------------------------------
## likfit: end of numerical maximisation.
## xvalid: number of data locations = 181
## xvalid: number of validation locations = 181
## xvalid: performing cross-validation at location ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181,
## xvalid: end of cross-validation## variog: computing omnidirectional variogram
## variofit: covariance model used is matern
## variofit: weights used: npairs
## variofit: minimisation function used: optim## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "14.83" "15.38" "14.83" "3"
## status "est" "est" "est" "fix"
## loss value: 20820.3835529824
## ---------------------------------------------------------------
## likfit: likelihood maximisation using the function optim.
## likfit: Use control() to pass additional
## arguments for the maximisation function.
## For further details see documentation for optim.
## likfit: It is highly advisable to run this function several
## times with different initial values for the parameters.
## likfit: WARNING: This step can be time demanding!
## ---------------------------------------------------------------
## likfit: end of numerical maximisation.
## xvalid: number of data locations = 173
## xvalid: number of validation locations = 173
## xvalid: performing cross-validation at location ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173,
## xvalid: end of cross-validation## variog: computing omnidirectional variogram
## variofit: covariance model used is exponential
## variofit: weights used: npairs
## variofit: minimisation function used: optim## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "0.01" "30.77" "0" "0.5"
## status "est" "est" "est" "fix"
## loss value: 0.00627202679678718
## kappa not used for the exponential correlation function
## ---------------------------------------------------------------
## likfit: likelihood maximisation using the function optim.
## likfit: Use control() to pass additional
## arguments for the maximisation function.
## For further details see documentation for optim.
## likfit: It is highly advisable to run this function several
## times with different initial values for the parameters.
## likfit: WARNING: This step can be time demanding!
## ---------------------------------------------------------------
## likfit: end of numerical maximisation.
## xvalid: number of data locations = 181
## xvalid: number of validation locations = 181
## xvalid: performing cross-validation at location ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181,
## xvalid: end of cross-validation## krige.conv: model with constant mean
## krige.conv: performing the Box-Cox data transformation
## krige.conv: back-transforming the predicted mean and variance
## krige.conv: back-transforming by simulating from the predictive.
## (run the function a few times and check stability of the results.
## krige.conv: Kriging performed using global neighbourhood
## variog: computing omnidirectional variogram
## variofit: covariance model used is exponential
## variofit: weights used: npairs
## variofit: minimisation function used: optim## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "2.04" "46.15" "0.68" "0.5"
## status "est" "est" "est" "fix"
## loss value: 226.407950874535
## kappa not used for the exponential correlation function
## ---------------------------------------------------------------
## likfit: likelihood maximisation using the function optim.
## likfit: Use control() to pass additional
## arguments for the maximisation function.
## For further details see documentation for optim.
## likfit: It is highly advisable to run this function several
## times with different initial values for the parameters.
## likfit: WARNING: This step can be time demanding!
## ---------------------------------------------------------------
## likfit: end of numerical maximisation.
## xvalid: number of data locations = 181
## xvalid: number of validation locations = 181
## xvalid: performing cross-validation at location ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181,
## xvalid: end of cross-validation## krige.conv: model with constant mean
## krige.conv: performing the Box-Cox data transformation
## krige.conv: back-transforming the predicted mean and variance
## krige.conv: back-transforming by simulating from the predictive.
## (run the function a few times and check stability of the results.
## krige.conv: Kriging performed using global neighbourhood## variog: computing omnidirectional variogram
## variofit: covariance model used is exponential
## variofit: weights used: npairs
## variofit: minimisation function used: optim## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "22.25" "30.77" "7.42" "0.5"
## status "est" "est" "est" "fix"
## loss value: 23905.6158470364
## kappa not used for the exponential correlation function
## ---------------------------------------------------------------
## likfit: likelihood maximisation using the function optim.
## likfit: Use control() to pass additional
## arguments for the maximisation function.
## For further details see documentation for optim.
## likfit: It is highly advisable to run this function several
## times with different initial values for the parameters.
## likfit: WARNING: This step can be time demanding!
## ---------------------------------------------------------------
## likfit: end of numerical maximisation.
## xvalid: number of data locations = 173
## xvalid: number of validation locations = 173
## xvalid: performing cross-validation at location ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173,
## xvalid: end of cross-validation## krige.conv: model with constant mean
## krige.conv: performing the Box-Cox data transformation
## krige.conv: back-transforming the predicted mean and variance
## krige.conv: back-transforming by simulating from the predictive.
## (run the function a few times and check stability of the results.
## krige.conv: Kriging performed using global neighbourhood## variog: computing omnidirectional variogram
## variofit: covariance model used is spherical
## variofit: weights used: npairs
## variofit: minimisation function used: optim## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "0.01" "76.92" "0" "0.5"
## status "est" "est" "est" "fix"
## loss value: 0.00949080358628735
## kappa not used for the spherical correlation function
## ---------------------------------------------------------------
## likfit: likelihood maximisation using the function optim.
## likfit: Use control() to pass additional
## arguments for the maximisation function.
## For further details see documentation for optim.
## likfit: It is highly advisable to run this function several
## times with different initial values for the parameters.
## likfit: WARNING: This step can be time demanding!
## ---------------------------------------------------------------
## likfit: end of numerical maximisation.
## xvalid: number of data locations = 181
## xvalid: number of validation locations = 181
## xvalid: performing cross-validation at location ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181,
## xvalid: end of cross-validation## krige.conv: model with constant mean
## krige.conv: performing the Box-Cox data transformation
## krige.conv: back-transforming the predicted mean and variance
## krige.conv: back-transforming by simulating from the predictive.
## (run the function a few times and check stability of the results.
## krige.conv: Kriging performed using global neighbourhood## variog: computing omnidirectional variogram
## variofit: covariance model used is spherical
## variofit: weights used: npairs
## variofit: minimisation function used: optim## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "2.04" "61.53" "0.27" "0.5"
## status "est" "est" "est" "fix"
## loss value: 253.742683880423
## kappa not used for the spherical correlation function
## ---------------------------------------------------------------
## likfit: likelihood maximisation using the function optim.
## likfit: Use control() to pass additional
## arguments for the maximisation function.
## For further details see documentation for optim.
## likfit: It is highly advisable to run this function several
## times with different initial values for the parameters.
## likfit: WARNING: This step can be time demanding!
## ---------------------------------------------------------------
## likfit: end of numerical maximisation.
## xvalid: number of data locations = 181
## xvalid: number of validation locations = 181
## xvalid: performing cross-validation at location ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181,
## xvalid: end of cross-validation## krige.conv: model with constant mean
## krige.conv: performing the Box-Cox data transformation
## krige.conv: back-transforming the predicted mean and variance
## krige.conv: back-transforming by simulating from the predictive.
## (run the function a few times and check stability of the results.
## krige.conv: Kriging performed using global neighbourhood
## variog: computing omnidirectional variogram
## variofit: covariance model used is spherical
## variofit: weights used: npairs
## variofit: minimisation function used: optim## variofit: searching for best initial value ... selected values:
## sigmasq phi tausq kappa
## initial.value "22.25" "76.92" "7.42" "0.5"
## status "est" "est" "est" "fix"
## loss value: 47549.2852145034
## kappa not used for the spherical correlation function
## ---------------------------------------------------------------
## likfit: likelihood maximisation using the function optim.
## likfit: Use control() to pass additional
## arguments for the maximisation function.
## For further details see documentation for optim.
## likfit: It is highly advisable to run this function several
## times with different initial values for the parameters.
## likfit: WARNING: This step can be time demanding!
## ---------------------------------------------------------------
## likfit: end of numerical maximisation.
## xvalid: number of data locations = 173
## xvalid: number of validation locations = 173
## xvalid: performing cross-validation at location ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173,
## xvalid: end of cross-validation## krige.conv: model with constant mean
## krige.conv: performing the Box-Cox data transformation
## krige.conv: back-transforming the predicted mean and variance
## krige.conv: back-transforming by simulating from the predictive.
## (run the function a few times and check stability of the results.
## krige.conv: Kriging performed using global neighbourhood## sph_error sph_cv exp_error exp_cv matern_error matern_cv
## 1976 NaN NaN NaN NaN NaN NaN
## 1979 NaN NaN NaN NaN NaN NaN
## 1982 NaN NaN NaN NaN NaN NaN## [1] "kappa=5"## 1976 lowest value:
## 1979 lowest value:
## 1982 lowest value:
Given this information, we proceed to use their parameters to estimate our beta values for each year, as well as the kriging of our predictions and the estimation of their confidence interval.
This concludes our estimation of each year of rainfall data using the ML method.
Comments
Overall we can see a general pattern of rainfall present around the western coast of Basc and Cal. This pattern is consistent across all years and could point to a proper general estimation of where rain occurs in these regions. Additionally, these patterns remain consistent even after estimating the upper and lower bounds, as well as generally following the original pattern of the data (as intended).
There are still concerning aspects of our estimation. Our errors seem to be relatively high, and while our rainfall estimates remain consistent with a somewhat stable spatial distribution, our variance gives us very wide confidence intervals, with hardly enough additional information added by our model.
Below we can see some quality measurements of our final estimation, particularly the ratio between the variance of our prediction and the variance of our data, as well as the median value of the relative width of each point.
Besides selecting our best model based on meassurement error, we can see by our coefficient of variation that our models are generally well defined, whith values close to 1, for the year 1976 the relationship betwene the model’s root mean square error and variation coefficient are seemingly inverse, which could suggest exploratory analysis on matern or spherical models for that year for even better results, however, the presently used model already has the best overall variation reduction and signficance levels.
Our worst estimate seems to be 1982, which also has a particularly explosive directional empirical variogram, and our second worst estimate is 1979, which has a very unusual growth pattern in the empirical variogram for the first few kilometers.
Overall our estimates are “good enough” in the sense that they provide minimally useful information, but they could be improved (perhaps creative use of covariates across years, algorithmic exploration of kappa values for the matern model, or dynamic cross validation with multiple parameters, as well as Bayesian estimation).